Optimal controller design for inverted pendulum system based on LQR method

This paper proposes design of optimal controller for inverted pendulum system. The main aim is to control angle of the pendulum and position of the cart simultaneously. Jacobian linearization technique has been used to linearize the system. Linear Quadratic Regulator technique is used here to solve the control problem. The solution of the Riccati is obtained by choosing appropriate weight matrix of continuous algebraic Riccati equation (CARE). The proposed controller able to meet the given performance specification minimizing the cost function efficiently. Finally, the performances of the controller have been evaluated with the presence of disturbances and variable initial conditions.

I Introduction

Inverted pendulum is a common non-linear system used in control theory to analyze the effectiveness of various control algorithms. The LQR method has been appeared several times in the literature to design optimal controller for inverted pendulum. Yong Xin et al.[1] proposed an approach of LQR based optimal controller for pendulum system considering initial angle very small. This approximation may cause a deviation between the theoretical model and the real system. This paper uses Jacobi linearization technique [2] for this nonlinear system which is being linearized around the equilibrium point. Linear Quadratic Regulator (LQR) solves the Riccati equation [2], [3] of a linear system that gives the Riccati solution I which can be obtained by choosing weighting matrix arbitrarily and keeping fixed weighting factor . This solution is used to construct a linear state-feedback optimal controller I minimizing the performance index of the system. The Inverted pendulum consists of two equilibrium points [4], one of them is stable while the other is unstable. The stable equilibrium corresponds to a state in which the pendulum is pointing downwards.